\(\int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx\) [11]

Optimal result

Integrand size = 29, antiderivative size = 210 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx=\frac {(B c-A d) \left (c^2-d^2 x^2\right )^{3/2}}{11 c d^2 (c+d x)^7}-\frac {(7 B c+4 A d) \left (c^2-d^2 x^2\right )^{3/2}}{99 c^2 d^2 (c+d x)^6}-\frac {(7 B c+4 A d) \left (c^2-d^2 x^2\right )^{3/2}}{231 c^3 d^2 (c+d x)^5}-\frac {2 (7 B c+4 A d) \left (c^2-d^2 x^2\right )^{3/2}}{1155 c^4 d^2 (c+d x)^4}-\frac {2 (7 B c+4 A d) \left (c^2-d^2 x^2\right )^{3/2}}{3465 c^5 d^2 (c+d x)^3} \] Output:

1/11*(-A*d+B*c)*(-d^2*x^2+c^2)^(3/2)/c/d^2/(d*x+c)^7-1/99*(4*A*d+7*B*c)*(- 
d^2*x^2+c^2)^(3/2)/c^2/d^2/(d*x+c)^6-1/231*(4*A*d+7*B*c)*(-d^2*x^2+c^2)^(3 
/2)/c^3/d^2/(d*x+c)^5-2/1155*(4*A*d+7*B*c)*(-d^2*x^2+c^2)^(3/2)/c^4/d^2/(d 
*x+c)^4-2/3465*(4*A*d+7*B*c)*(-d^2*x^2+c^2)^(3/2)/c^5/d^2/(d*x+c)^3
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.61 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx=-\frac {(c-d x) \sqrt {c^2-d^2 x^2} \left (7 B c \left (13 c^4+91 c^3 d x+45 c^2 d^2 x^2+14 c d^3 x^3+2 d^4 x^4\right )+A d \left (547 c^4+364 c^3 d x+180 c^2 d^2 x^2+56 c d^3 x^3+8 d^4 x^4\right )\right )}{3465 c^5 d^2 (c+d x)^6} \] Input:

Integrate[((A + B*x)*Sqrt[c^2 - d^2*x^2])/(c + d*x)^7,x]
 

Output:

-1/3465*((c - d*x)*Sqrt[c^2 - d^2*x^2]*(7*B*c*(13*c^4 + 91*c^3*d*x + 45*c^ 
2*d^2*x^2 + 14*c*d^3*x^3 + 2*d^4*x^4) + A*d*(547*c^4 + 364*c^3*d*x + 180*c 
^2*d^2*x^2 + 56*c*d^3*x^3 + 8*d^4*x^4)))/(c^5*d^2*(c + d*x)^6)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {671, 461, 461, 461, 460}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx\)

\(\Big \downarrow \) 671

\(\displaystyle \frac {(4 A d+7 B c) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^6}dx}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{3/2} (B c-A d)}{11 c d^2 (c+d x)^7}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+7 B c) \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^5}dx}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{3/2} (B c-A d)}{11 c d^2 (c+d x)^7}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+7 B c) \left (\frac {\frac {2 \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^4}dx}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{3/2} (B c-A d)}{11 c d^2 (c+d x)^7}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {(4 A d+7 B c) \left (\frac {\frac {2 \left (\frac {\int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^3}dx}{5 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right )}{11 c d}+\frac {\left (c^2-d^2 x^2\right )^{3/2} (B c-A d)}{11 c d^2 (c+d x)^7}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{3/2} (B c-A d)}{11 c d^2 (c+d x)^7}+\frac {\left (\frac {\frac {2 \left (-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{15 c^2 d (c+d x)^3}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{5 c d (c+d x)^4}\right )}{7 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{7 c d (c+d x)^5}}{3 c}-\frac {\left (c^2-d^2 x^2\right )^{3/2}}{9 c d (c+d x)^6}\right ) (4 A d+7 B c)}{11 c d}\)

Input:

Int[((A + B*x)*Sqrt[c^2 - d^2*x^2])/(c + d*x)^7,x]
 

Output:

((B*c - A*d)*(c^2 - d^2*x^2)^(3/2))/(11*c*d^2*(c + d*x)^7) + ((7*B*c + 4*A 
*d)*(-1/9*(c^2 - d^2*x^2)^(3/2)/(c*d*(c + d*x)^6) + (-1/7*(c^2 - d^2*x^2)^ 
(3/2)/(c*d*(c + d*x)^5) + (2*(-1/5*(c^2 - d^2*x^2)^(3/2)/(c*d*(c + d*x)^4) 
 - (c^2 - d^2*x^2)^(3/2)/(15*c^2*d*(c + d*x)^3)))/(7*c))/(3*c)))/(11*c*d)
 

Defintions of rubi rules used

rule 460

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, 
 p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
 

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m 
 + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m 
+ p + 1))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, 
e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p 
 + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p 
 + 1, 0]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63

method	result	size
gosper	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{5} x^{4}+14 B c \,d^{4} x^{4}+56 A c \,d^{4} x^{3}+98 B \,c^{2} d^{3} x^{3}+180 A \,c^{2} d^{3} x^{2}+315 B \,c^{3} d^{2} x^{2}+364 A \,c^{3} d^{2} x +637 B \,c^{4} d x +547 A \,c^{4} d +91 B \,c^{5}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 \left (d x +c \right )^{6} c^{5} d^{2}}\)	\(133\)
orering	\(-\frac {\left (-d x +c \right ) \left (8 A \,d^{5} x^{4}+14 B c \,d^{4} x^{4}+56 A c \,d^{4} x^{3}+98 B \,c^{2} d^{3} x^{3}+180 A \,c^{2} d^{3} x^{2}+315 B \,c^{3} d^{2} x^{2}+364 A \,c^{3} d^{2} x +637 B \,c^{4} d x +547 A \,c^{4} d +91 B \,c^{5}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 \left (d x +c \right )^{6} c^{5} d^{2}}\)	\(133\)
trager	\(-\frac {\left (-8 A \,d^{6} x^{5}-14 B c \,d^{5} x^{5}-48 A c \,d^{5} x^{4}-84 B \,c^{2} d^{4} x^{4}-124 A \,c^{2} d^{4} x^{3}-217 B \,c^{3} d^{3} x^{3}-184 A \,c^{3} d^{3} x^{2}-322 B \,c^{4} d^{2} x^{2}-183 A \,c^{4} d^{2} x +546 B \,c^{5} d x +547 A \,c^{5} d +91 B \,c^{6}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{3465 c^{5} \left (d x +c \right )^{6} d^{2}}\)	\(151\)
default	\(\frac {B \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{6}}+\frac {d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{5}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{5 c d \left (x +\frac {c}{d}\right )^{4}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{15 c^{2} \left (x +\frac {c}{d}\right )^{3}}\right )}{7 c}\right )}{3 c}\right )}{d^{7}}+\frac {\left (A d -B c \right ) \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{11 c d \left (x +\frac {c}{d}\right )^{7}}+\frac {4 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{9 c d \left (x +\frac {c}{d}\right )^{6}}+\frac {d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{7 c d \left (x +\frac {c}{d}\right )^{5}}+\frac {2 d \left (-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{5 c d \left (x +\frac {c}{d}\right )^{4}}-\frac {\left (-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )\right )^{\frac {3}{2}}}{15 c^{2} \left (x +\frac {c}{d}\right )^{3}}\right )}{7 c}\right )}{3 c}\right )}{11 c}\right )}{d^{8}}\)	\(455\)

Input:

int((B*x+A)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^7,x,method=_RETURNVERBOSE)
 

Output:

-1/3465*(-d*x+c)*(8*A*d^5*x^4+14*B*c*d^4*x^4+56*A*c*d^4*x^3+98*B*c^2*d^3*x 
^3+180*A*c^2*d^3*x^2+315*B*c^3*d^2*x^2+364*A*c^3*d^2*x+637*B*c^4*d*x+547*A 
*c^4*d+91*B*c^5)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^6/c^5/d^2
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx=-\frac {91 \, B c^{7} + 547 \, A c^{6} d + {\left (91 \, B c d^{6} + 547 \, A d^{7}\right )} x^{6} + 6 \, {\left (91 \, B c^{2} d^{5} + 547 \, A c d^{6}\right )} x^{5} + 15 \, {\left (91 \, B c^{3} d^{4} + 547 \, A c^{2} d^{5}\right )} x^{4} + 20 \, {\left (91 \, B c^{4} d^{3} + 547 \, A c^{3} d^{4}\right )} x^{3} + 15 \, {\left (91 \, B c^{5} d^{2} + 547 \, A c^{4} d^{3}\right )} x^{2} + 6 \, {\left (91 \, B c^{6} d + 547 \, A c^{5} d^{2}\right )} x + {\left (91 \, B c^{6} + 547 \, A c^{5} d - 2 \, {\left (7 \, B c d^{5} + 4 \, A d^{6}\right )} x^{5} - 12 \, {\left (7 \, B c^{2} d^{4} + 4 \, A c d^{5}\right )} x^{4} - 31 \, {\left (7 \, B c^{3} d^{3} + 4 \, A c^{2} d^{4}\right )} x^{3} - 46 \, {\left (7 \, B c^{4} d^{2} + 4 \, A c^{3} d^{3}\right )} x^{2} + 3 \, {\left (182 \, B c^{5} d - 61 \, A c^{4} d^{2}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{3465 \, {\left (c^{5} d^{8} x^{6} + 6 \, c^{6} d^{7} x^{5} + 15 \, c^{7} d^{6} x^{4} + 20 \, c^{8} d^{5} x^{3} + 15 \, c^{9} d^{4} x^{2} + 6 \, c^{10} d^{3} x + c^{11} d^{2}\right )}} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^7,x, algorithm="fricas")
 

Output:

-1/3465*(91*B*c^7 + 547*A*c^6*d + (91*B*c*d^6 + 547*A*d^7)*x^6 + 6*(91*B*c 
^2*d^5 + 547*A*c*d^6)*x^5 + 15*(91*B*c^3*d^4 + 547*A*c^2*d^5)*x^4 + 20*(91 
*B*c^4*d^3 + 547*A*c^3*d^4)*x^3 + 15*(91*B*c^5*d^2 + 547*A*c^4*d^3)*x^2 + 
6*(91*B*c^6*d + 547*A*c^5*d^2)*x + (91*B*c^6 + 547*A*c^5*d - 2*(7*B*c*d^5 
+ 4*A*d^6)*x^5 - 12*(7*B*c^2*d^4 + 4*A*c*d^5)*x^4 - 31*(7*B*c^3*d^3 + 4*A* 
c^2*d^4)*x^3 - 46*(7*B*c^4*d^2 + 4*A*c^3*d^3)*x^2 + 3*(182*B*c^5*d - 61*A* 
c^4*d^2)*x)*sqrt(-d^2*x^2 + c^2))/(c^5*d^8*x^6 + 6*c^6*d^7*x^5 + 15*c^7*d^ 
6*x^4 + 20*c^8*d^5*x^3 + 15*c^9*d^4*x^2 + 6*c^10*d^3*x + c^11*d^2)
 

Sympy [F]

\[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx=\int \frac {\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \left (A + B x\right )}{\left (c + d x\right )^{7}}\, dx \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**(1/2)/(d*x+c)**7,x)
 

Output:

Integral(sqrt(-(-c + d*x)*(c + d*x))*(A + B*x)/(c + d*x)**7, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (190) = 380\).

Time = 0.06 (sec) , antiderivative size = 1009, normalized size of antiderivative = 4.80 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^7,x, algorithm="maxima")
 

Output:

2/11*sqrt(-d^2*x^2 + c^2)*B*c/(d^8*x^6 + 6*c*d^7*x^5 + 15*c^2*d^6*x^4 + 20 
*c^3*d^5*x^3 + 15*c^4*d^4*x^2 + 6*c^5*d^3*x + c^6*d^2) - 1/99*sqrt(-d^2*x^ 
2 + c^2)*B*c/(c*d^7*x^5 + 5*c^2*d^6*x^4 + 10*c^3*d^5*x^3 + 10*c^4*d^4*x^2 
+ 5*c^5*d^3*x + c^6*d^2) - 4/693*sqrt(-d^2*x^2 + c^2)*B*c/(c^2*d^6*x^4 + 4 
*c^3*d^5*x^3 + 6*c^4*d^4*x^2 + 4*c^5*d^3*x + c^6*d^2) - 4/1155*sqrt(-d^2*x 
^2 + c^2)*B*c/(c^3*d^5*x^3 + 3*c^4*d^4*x^2 + 3*c^5*d^3*x + c^6*d^2) - 8/34 
65*sqrt(-d^2*x^2 + c^2)*B*c/(c^4*d^4*x^2 + 2*c^5*d^3*x + c^6*d^2) - 8/3465 
*sqrt(-d^2*x^2 + c^2)*B*c/(c^5*d^3*x + c^6*d^2) - 2/11*sqrt(-d^2*x^2 + c^2 
)*A/(d^7*x^6 + 6*c*d^6*x^5 + 15*c^2*d^5*x^4 + 20*c^3*d^4*x^3 + 15*c^4*d^3* 
x^2 + 6*c^5*d^2*x + c^6*d) + 1/99*sqrt(-d^2*x^2 + c^2)*A/(c*d^6*x^5 + 5*c^ 
2*d^5*x^4 + 10*c^3*d^4*x^3 + 10*c^4*d^3*x^2 + 5*c^5*d^2*x + c^6*d) + 4/693 
*sqrt(-d^2*x^2 + c^2)*A/(c^2*d^5*x^4 + 4*c^3*d^4*x^3 + 6*c^4*d^3*x^2 + 4*c 
^5*d^2*x + c^6*d) + 4/1155*sqrt(-d^2*x^2 + c^2)*A/(c^3*d^4*x^3 + 3*c^4*d^3 
*x^2 + 3*c^5*d^2*x + c^6*d) + 8/3465*sqrt(-d^2*x^2 + c^2)*A/(c^4*d^3*x^2 + 
 2*c^5*d^2*x + c^6*d) + 8/3465*sqrt(-d^2*x^2 + c^2)*A/(c^5*d^2*x + c^6*d) 
- 2/9*sqrt(-d^2*x^2 + c^2)*B/(d^7*x^5 + 5*c*d^6*x^4 + 10*c^2*d^5*x^3 + 10* 
c^3*d^4*x^2 + 5*c^4*d^3*x + c^5*d^2) + 1/63*sqrt(-d^2*x^2 + c^2)*B/(c*d^6* 
x^4 + 4*c^2*d^5*x^3 + 6*c^3*d^4*x^2 + 4*c^4*d^3*x + c^5*d^2) + 1/105*sqrt( 
-d^2*x^2 + c^2)*B/(c^2*d^5*x^3 + 3*c^3*d^4*x^2 + 3*c^4*d^3*x + c^5*d^2) + 
2/315*sqrt(-d^2*x^2 + c^2)*B/(c^3*d^4*x^2 + 2*c^4*d^3*x + c^5*d^2) + 2/...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (190) = 380\).

Time = 0.14 (sec) , antiderivative size = 666, normalized size of antiderivative = 3.17 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^7,x, algorithm="giac")
 

Output:

2/3465*(91*B*c + 547*A*d + 1001*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*B*c/(d 
^2*x) + 2552*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*A/(d*x) + 1540*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))^2*B*c/(d^4*x^2) + 16225*(c*d + sqrt(-d^2*x^2 + c 
^2)*abs(d))^2*A/(d^3*x^2) + 9240*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*B*c 
/(d^6*x^3) + 42900*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*A/(d^5*x^3) + 115 
50*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*B*c/(d^8*x^4) + 92730*(c*d + sqrt 
(-d^2*x^2 + c^2)*abs(d))^4*A/(d^7*x^4) + 24486*(c*d + sqrt(-d^2*x^2 + c^2) 
*abs(d))^5*B*c/(d^10*x^5) + 122892*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^5*A 
/(d^9*x^5) + 17556*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*B*c/(d^12*x^6) + 
129822*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^6*A/(d^11*x^6) + 18480*(c*d + s 
qrt(-d^2*x^2 + c^2)*abs(d))^7*B*c/(d^14*x^7) + 87780*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^7*A/(d^13*x^7) + 5775*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8 
*B*c/(d^16*x^8) + 47355*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^8*A/(d^15*x^8) 
 + 3465*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^9*B*c/(d^18*x^9) + 13860*(c*d 
+ sqrt(-d^2*x^2 + c^2)*abs(d))^9*A/(d^17*x^9) + 3465*(c*d + sqrt(-d^2*x^2 
+ c^2)*abs(d))^10*A/(d^19*x^10))/(c^5*d*((c*d + sqrt(-d^2*x^2 + c^2)*abs(d 
))/(d^2*x) + 1)^11*abs(d))
 

Mupad [B] (verification not implemented)

Time = 10.87 (sec) , antiderivative size = 1141, normalized size of antiderivative = 5.43 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx =\text {Too large to display} \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(A + B*x))/(c + d*x)^7,x)
 

Output:

((c^2 - d^2*x^2)^(1/2)*(B/(495*c^2*d^2) - (4*A*d - 60*B*c)/(3465*c^3*d^2)) 
)/(c + d*x)^3 - ((c^2 - d^2*x^2)^(1/2)*(B/(77*c*d^2) - (5*A*d - 15*B*c)/(6 
93*c^2*d^2)))/(c + d*x)^4 - ((c^2 - d^2*x^2)^(1/2)*(B/(2079*c^3*d^2) - (3* 
A*d - 75*B*c)/(10395*c^4*d^2)))/(c + d*x)^2 + ((c^2 - d^2*x^2)^(1/2)*(B/(3 
465*c^4*d^2) - (2*A*d - 66*B*c)/(10395*c^5*d^2)))/(c + d*x) - ((c^2 - d^2* 
x^2)^(1/2)*(A/(11*d) - (c*(B/(11*d) - (A*d - B*c)/(11*c*d)))/d))/(c + d*x) 
^6 - ((c^2 - d^2*x^2)^(1/2)*((12*B*c^2 - A*c*d)/(99*c^2*d^2) - (c*((A*d^2 
- 13*B*c*d)/(99*c^2*d^2) - B/(99*c*d)))/d))/(c + d*x)^5 + ((c^2 - d^2*x^2) 
^(1/2)*((22*B*c^2 - A*c*d)/(693*c^3*d^2) - (c*((A*d^2 - 23*B*c*d)/(693*c^3 
*d^2) - B/(693*c^2*d)))/d))/(c + d*x)^4 - ((c^2 - d^2*x^2)^(1/2)*((30*B*c^ 
2 - A*c*d)/(3465*c^4*d^2) - (c*((A*d^2 - 31*B*c*d)/(3465*c^4*d^2) - B/(346 
5*c^3*d)))/d))/(c + d*x)^3 + ((c^2 - d^2*x^2)^(1/2)*((36*B*c^2 - A*c*d)/(1 
0395*c^5*d^2) - (c*((A*d^2 - 37*B*c*d)/(10395*c^5*d^2) - B/(10395*c^4*d))) 
/d))/(c + d*x)^2 - ((c^2 - d^2*x^2)^(1/2)*((40*B*c^2 - A*c*d)/(10395*c^6*d 
^2) - (c*((A*d^2 - 41*B*c*d)/(10395*c^6*d^2) - B/(10395*c^5*d)))/d))/(c + 
d*x) + ((c^2 - d^2*x^2)^(1/2)*(5*A*d - 3*B*c))/(1155*c^3*d^2*(c + d*x)^3) 
+ ((c^2 - d^2*x^2)^(1/2)*(10*A*d - 6*B*c))/(3465*c^4*d^2*(c + d*x)^2) + (( 
c^2 - d^2*x^2)^(1/2)*(10*A*d - 6*B*c))/(3465*c^5*d^2*(c + d*x)) + ((c^2 - 
d^2*x^2)^(1/2)*(3*A*d - 65*B*c))/(10395*c^5*d^2*(c + d*x)) - ((c^2 - d^2*x 
^2)^(1/2)*(8*A*d - 99*B*c))/(10395*c^4*d^2*(c + d*x)^2) - ((c^2 - d^2*x...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.78 \[ \int \frac {(A+B x) \sqrt {c^2-d^2 x^2}}{(c+d x)^7} \, dx=\frac {630 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5}+232 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d x +646 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{2} x^{2}+706 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{3} x^{3}+367 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{4} x^{4}+75 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{5} x^{5}+91 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} x -1232 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d \,x^{2}-1127 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{2} x^{3}-539 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{3} x^{4}-105 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{4} x^{5}-630 a \,c^{6}+232 a \,c^{5} d x -1244 a \,c^{4} d^{2} x^{2}-1720 a \,c^{3} d^{3} x^{3}-1321 a \,c^{2} d^{4} x^{4}-538 a c \,d^{5} x^{5}-91 a \,d^{6} x^{6}+91 b \,c^{6} x +2233 b \,c^{5} d \,x^{2}+1715 b \,c^{4} d^{2} x^{3}+1232 b \,c^{3} d^{3} x^{4}+476 b \,c^{2} d^{4} x^{5}+77 b c \,d^{5} x^{6}}{3465 c^{5} d \left (\sqrt {-d^{2} x^{2}+c^{2}}\, c^{5}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d x +10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+10 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+5 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+\sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}-c^{6}-6 c^{5} d x -15 c^{4} d^{2} x^{2}-20 c^{3} d^{3} x^{3}-15 c^{2} d^{4} x^{4}-6 c \,d^{5} x^{5}-d^{6} x^{6}\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^(1/2)/(d*x+c)^7,x)
 

Output:

(630*sqrt(c**2 - d**2*x**2)*a*c**5 + 232*sqrt(c**2 - d**2*x**2)*a*c**4*d*x 
 + 646*sqrt(c**2 - d**2*x**2)*a*c**3*d**2*x**2 + 706*sqrt(c**2 - d**2*x**2 
)*a*c**2*d**3*x**3 + 367*sqrt(c**2 - d**2*x**2)*a*c*d**4*x**4 + 75*sqrt(c* 
*2 - d**2*x**2)*a*d**5*x**5 + 91*sqrt(c**2 - d**2*x**2)*b*c**5*x - 1232*sq 
rt(c**2 - d**2*x**2)*b*c**4*d*x**2 - 1127*sqrt(c**2 - d**2*x**2)*b*c**3*d* 
*2*x**3 - 539*sqrt(c**2 - d**2*x**2)*b*c**2*d**3*x**4 - 105*sqrt(c**2 - d* 
*2*x**2)*b*c*d**4*x**5 - 630*a*c**6 + 232*a*c**5*d*x - 1244*a*c**4*d**2*x* 
*2 - 1720*a*c**3*d**3*x**3 - 1321*a*c**2*d**4*x**4 - 538*a*c*d**5*x**5 - 9 
1*a*d**6*x**6 + 91*b*c**6*x + 2233*b*c**5*d*x**2 + 1715*b*c**4*d**2*x**3 + 
 1232*b*c**3*d**3*x**4 + 476*b*c**2*d**4*x**5 + 77*b*c*d**5*x**6)/(3465*c* 
*5*d*(sqrt(c**2 - d**2*x**2)*c**5 + 5*sqrt(c**2 - d**2*x**2)*c**4*d*x + 10 
*sqrt(c**2 - d**2*x**2)*c**3*d**2*x**2 + 10*sqrt(c**2 - d**2*x**2)*c**2*d* 
*3*x**3 + 5*sqrt(c**2 - d**2*x**2)*c*d**4*x**4 + sqrt(c**2 - d**2*x**2)*d* 
*5*x**5 - c**6 - 6*c**5*d*x - 15*c**4*d**2*x**2 - 20*c**3*d**3*x**3 - 15*c 
**2*d**4*x**4 - 6*c*d**5*x**5 - d**6*x**6))